There is basically no way to make progress here, as far as I can see. If the insurance companies weren't running open-loop before, they certainly are now.
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My doctor wanted to give me an MRI for a pain near the heart, and insurance told them they wouldn't cover it until they did various other forms of cheaper treatment, including taking antacids for one month, and 5 months of physical therapy. Which of course didn't work. The waiting time for the first appointment was 3 months.
It took over 9 months for my doctor, the only person to actually properly know the details of my case, to be able to give an MRI that he thought was necessary because someone at the insurance company, who I never met, who had less medical expertise than my doctor, wanted to save the insurance company money.
Anecdotally, all the people I know who live in countries with socialized medicine haven't ever had a wait time as long as that, and haven't ever had a simple MRI be delayed by their socialized insurance.
There are already large swaths of people who can't get a MRI period because they are excluded from healthcare in this country. And if your belief is that a single payer system will be hampered by measures such as austerity, then say that. Because then your issue isn't with single payer healthcare but instead with politicians who believe institutions need to be run as a business instead of what they actually are, which are public services.
Australia has a combination of public and private health insurance, and they both work well together. The public health options provide the safety net, while the private health insurance is optional.
Where the private option makes sense is if you want to go to specific private hospitals, or if you have elective surgery (the classic example being a knee reconstruction for sports injury) and you don't want to be in a queue behind people waiting for public hospital beds for more serious conditions like heart surgery and so on.
My dad in Australia had open heart surgery 2 years ago, and is doing very well. His cost for the entire procedure? $0 and this was done by one of the very best heart surgeons in the country. He has private health insurance, but elects to go to public hospitals, which have excellent surgeons committed to the best care, because he's a patriotic sort and he's paid into the public health system through taxes for his entire life.
Meanwhile I pay > $3k per-month for not even top tier care in Upstate New York for myself, my wife and my 10 year old daughter, with no serious pre-existing conditions, and I have absolutely no guarantee that any surgery or anything that any of us need in the future will be even covered, even if my primary physician says it's medically necessary.
The rest of the world would do well to study how the combination of public and private health insurance is done in Australia.
Far better to instead wait for 11 months, spend a couple hundred hours bouncing around phone trees for my health insurance provider, and pay a few thousand dollars per month for having the privilege to do so. So much less stress for me and my doctors to get jerked around by barely medically competent insurance company employees, makes it much easier to sleep at night. Especially fun to have PA withdrawn because due to how long it's taken to get the MRI has allowed the issue to progress to the point where the kind of MRI needed is now different and requires a new PA.